Abstract
Random sequences with random or stochastic indices controlled by a doubly stochastic Poisson process are considered in this paper. A Poisson stochastic index process (PSI-process) is a random process with the continuous time ψ(t) obtained by subordinating a sequence of random variables (ξj), j = 0, 1, …, by a doubly stochastic Poisson process Π1(tλ) via the substitution ψ(t) = \({{\xi }_{{{{\Pi }_{1}}(t\lambda )}}}\), t \( \geqslant \) 0, where the random intensity λ is assumed independent of the standard Poisson process Π1. In this paper, we restrict our consideration to the case of independent identically distributed random variables (ξj) with a finite variance. We find a representation of the fractional Ornstein–Uhlenbeck process with the Hurst exponent H ∈ (0, 1/2) introduced and investigated by R. Wolpert and M. Taqqu (2005) in the form of a limit of normalized sums of independent identically distributed PSI-processes with an explicitly given distribution of the random intensity λ. This fractional Ornstein–Uhlenbeck process provides a local, at t = 0, mean-square approximation of the fractional Brownian motion with the same Hurst exponent H ∈ (0, 1/2). We examine in detail two examples of PSI-processes with the random intensity λ generating the fractional Ornstein–Uhlenbeck process in the Wolpert and Taqqu sense. These are a telegraph process arising when ξ0 has a Rademacher distribution ±1 with the probability 1/2 and a PSI-process with the uniform distribution for ξ0. For these two examples, we calculate the exact and the asymptotic values of the local modulus of continuity for a single PSI-process over a small fixed time span.
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Notes
Note that an integral written in the form of (8) can be equivalently defined as a stochastic integral over the increments of standard two-direction Brownian motion dW(s), s ∈ \(\mathbb{R}\), with the increments introduced formally into (8) instead of W(ds) (see, e.g., [13], Ch. IX).
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Funding
The contributions of O.V. Rusakov and Yu.V. Yakubovich were partially supported by the Russian Foundation for Basic Research, project no. 20-01-00646 A.
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Translated by N. Semenova
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Rusakov, O.V., Yakubovich, Y.V. & Baev, B.A. On Some Local Asymptotic Properties of Sequences with a Random Index. Vestnik St.Petersb. Univ.Math. 53, 308–319 (2020). https://doi.org/10.1134/S1063454120030115
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DOI: https://doi.org/10.1134/S1063454120030115